Topics
2. Exponents & Scientific Notation
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United States of America
Grade 8

2.04 Power of a power property

Lesson
Practice

Introduction

We previously learned about the product of powers property. We'll now look at how to find the power of a power and the corresponding property.

Power of a power property

Let's consider how we might rewrite expressions that have an exponential raised to another power.

Consider the expression (a^{2})^{3}. What is the resulting power of base a? To find out, have a look at the expanded form of the expression:

\displaystyle (a^{2})^{3}\displaystyle =\displaystyle (a^{2})\times (a^{2}) \times (a^{2})
\displaystyle =\displaystyle (a \times a) \times (a \times a) \times (a \times a)
\displaystyle =\displaystyle a \times a \times a \times a \times a \times a
\displaystyle =\displaystyle a^{6}

In the expanded form, we can see that we are multiplying six groups of a together. That is, \\(a^{2})^{3}=a^{6}.

We can confirm this result using the product of powers rule:

\displaystyle (a^{2}) \times (a^{2}) \times (a^{2})\displaystyle =\displaystyle a^{2+2+2}
\displaystyle =\displaystyle a^6

We can avoid having to write each expression in expanded form by using the power of a power law.

For any base number a, and any numbers m and n as power, (a^{m})^{n} = a^{m\times n}

That is, when simplifying a term with a power that itself has a power:

  • Keep the same base

  • Multiply the exponents

Examples

Example 1

We want to simplify: (2^{2})^{3}.

a

Select the three expressions which are equivalent to (2^{2})^{3}:

A
2^{2}\times 2^{3}
B
(2\times 2) \times ( 2\times 2\times 2)
C
(2 \times 2)^{3}
D
(2\times 2) \times (2\times 2) \times (2\times 2)
E
2^{2} \times 2^{2} \times 2^{2}
Worked Solution
Create a strategy

Choose options that have the same expanded form.

Apply the idea
\displaystyle (2^{2})^{3}\displaystyle =\displaystyle 2^2 \times 2^2 \times 2^2 Expand the power of 3
\displaystyle =\displaystyle 2 \times 2 \times 2 \times 2 \times 2 \times 2 Expand the squares

So any option with six 2's in expanded form are equivalent.

Option A: 2^{2} \times 2^{3} = 2 \times 2 \times 2 \times 2 \times 2

Option B: (2\times 2) \times (2\times 2\times 2) = 2 \times 2 \times 2 \times 2 \times 2

Option C:

\displaystyle (2 \times 2)^{3}\displaystyle =\displaystyle (2\times 2) \times (2\times 2) \times (2\times 2)
\displaystyle =\displaystyle 2 \times 2 \times 2 \times 2 \times 2 \times 2

Option D:

\displaystyle (2\times 2) \times (2\times 2) \times (2\times 2) \displaystyle =\displaystyle 2 \times 2 \times 2 \times 2 \times 2 \times 2

Option E:

\displaystyle 2^{2} \times 2^{2} \times 2^{2} \displaystyle =\displaystyle (2\times 2) \times (2\times 2) \times (2\times 2)
\displaystyle =\displaystyle 2 \times 2 \times 2 \times 2 \times 2 \times 2

The correct options are C, D, and E because they all have an expanded form with six 2's like (2^{2})^{3}.

b

Complete the rule: \left(2^{2}\right)^{3} = 2^{⬚}

Worked Solution
Create a strategy

We can use the power of power law: (a^{m})^{n}=a^{m \times n}

Apply the idea
\displaystyle \left(2^{2}\right)^{3}\displaystyle =\displaystyle 2^{2 \times 3}Multiply the powers
\displaystyle \left(2^{2}\right)^{3}\displaystyle =\displaystyle 2^{6}Evaluate the power
Idea summary

For any base number a, and any numbers m and n as power, (a^{m})^{n} = a^{m\times n}

That is, when simplifying a term with a power that itself has a power:

  • Keep the same base

  • Multiply the exponents

Outcomes

8.EE.A.1

Know and apply the properties of integer exponents to generate equivalent numerical expressions.

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